BOUNDING χ BY A FRACTION OF ∆ FOR GRAPHS WITHOUT LARGE CLIQUES

The greedy coloring algorithm shows that a graph of maximum degree at most ∆ has chromatic number at most ∆ + 1, and this is tight for cliques. Much attention has been devoted to improving this “greedy bound” for graphs without large cliques. Brooks famously proved that this bound can be improved by one if ∆ ≥ 3 and the graph contains no clique of size ∆ + 1. Reed’s Conjecture states that the “greedy bound” can be improved by k if the graph contains no clique of size ∆ + 1 − 2k. Johansson proved that the “greedy bound” can be improved by a factor of Ω(ln(∆)−1) or Ω (